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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">v</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">w</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b2">p</font></span>)</span> holds <br/>( <font color="Olive" title="b1">J</font>,<font color="Olive" title="b3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">p</font> iff <font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">w</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">p</font> )</span><br/><b>let </b><font color="Maroon" title="c2">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">v</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">w</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b1">p</font></span>)</span> holds <br/>( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b2">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p</font> iff <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b3">w</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p</font> )</span><br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ] <b>means </b>for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">v</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> $1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">w</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> $1</span>)</span> holds <br/>( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> $1 iff <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b2">w</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> $1 );<br/>
<a NAME="E1:87"/><i><font color="Green" title="E66">A1</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b3">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b4">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b5">l</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b4">k</font><br/> for <font color="Olive" title="b6">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b4">k</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b6">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b5">l</font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Olive" title="b1">p</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b2">q</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">p</font>] ) )
 
<b>by </b><i><a class="ref" href="sublemma.html#T61" target="_self" title="SUBLEMMA:th.61">Th61</a>, <a class="ref" href="sublemma.html#T62" target="_self" title="SUBLEMMA:th.62">Th62</a>, <a class="ref" href="sublemma.html#T63" target="_self" title="SUBLEMMA:th.63">Th63</a>, <a class="ref" href="sublemma.html#T68" target="_self" title="SUBLEMMA:th.68">Th68</a>, <a class="ref" href="valuat_1.html#T44" title="VALUAT_1:th.44">VALUAT_1:44</a></i>;<br/>
<b>thus </b><a NAME="E2:87"/>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 <b>from </b><i><a class="ref" href="cqc_lang.html#S1" title="CQC_LANG:sch.1">CQC_LANG:sch 1</a>(<a class="txt" href="#E1:87"><i><font color="Green" title="E66">A1</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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